Maybe this will clear it up for you: That the physicists are making a normal distribution assumption is from shaky down to sloppy down to silly down to absurd. Making this assumption for experimental data, e.g., the LHC data or the gravitational wave data, where they are hanging on by their fingernails anyway, is close to treating the normal distribution as some gift from God.

And, if some field makes a normal, or mean 0, variance 1 normal, assumption, then they should say so and, hopefully, justify that assumption.

Your examples from Wolfram and Wikipedia show that a lot of people have guzzled that old Kool Aid on the normal distribution. A few years ago I had a date with a high school teacher, and she assumed that test scores are normal. My father had an MS in education and my brother was a ugrad psych major, so, when my brother's psych material got to statistics, Dad taught him about the normal distribution. It appears that around 1900 and then for 80 years or so, and still in some parts of some fields, a normal assumption, without mention or justification, was common and remains in what you call the "context". "Context" or not, common or not, popular or not, it's still an assumption, needs mention and justification, and usually is not at all well justified or justifiable.

Sure, with their normal assumption and, say, five sigma, experimental physicists get particular probabilities of Type I error, so called p values. No question here.

The issue is the normal assumption.

There are places where a normal assumption is quite solid. The most common justification is from the central limit theorem. So, take random variables X(1), X(2), ... and assume that they are independent and all have the same distribution with finite standard deviation. Then the central limit theorem says essentially that as positive integer n grows to infinity the distribution of the sum

X(1) + ... + X(n)

will converge to normal (really should divide the sum by square root of n).

So, where might we get some such

X(1) + ... + X(n)?

Sure, from Brownian motion where there are many little bumps and the bumps come close enough to satisfying the hypothesis. So can use this also in thermodynamics.

So, there are places where a normal assumption is justified.

But just making a normal assumption for essentially all experimental errors as needed for the accepted five sigma criterion is close to a weird religion and not flattering to a modern science.

What physics is doing with their five sigma criterion is a statistical hypothesis test. My examples of distribution free hypothesis tests via resampling is a better justified and more conservative and robust way to do an hypothesis test. Physicists might consider using such.

Such distribution-free statistical methods form a large field.

My example of the role of Hilbert space is quite practical.

If you want to explain to the common man in the street why physics likes their five sigma criterion, then, sure, you need the normal assumption. Then you should mention that there physics is making a normal assumption.

There you might not confuse the common man in the street with a claim that in making this normal assumption physics is close to drinking some swill of boiled tails of rats and bats.

But with the now high interest in computer science of big data, machine learning, etc. I was assuming that the HN audience could and should hear the real stuff -- that in what physics is doing, there's a normal (Gaussian) assumption in there.